CoCalc Public Fileswww / projects / kleinerman_senior_thesis.tex
Author: William A. Stein
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101
102
103\begin{document}
104
105\title{On the Torsion Points of Elliptic Curves \& Modular Abelian Varieties}
106\author{Seth Kleinerman}
107\maketitle
108
109\tableofcontents
110\newpage
111\section{Introduction and Background}
112
113\begin{flushleft}
114\begin{verse}
115Age cannot wither her, nor custom stale\\
116Her infinite variety.\\
117---\emph{Antony and Cleopatra}, 2.2.271-2.\\
118\end{verse}
119\end{flushleft}
120
121The study of abelian varieties is of great interest in modern number theory. In this paper we will try to understand some of the basics of the varieties' torsion subgroups. That they are finite at all is a matter of some concern, and that will be the main thrust of the theory we develop. We will often specialize to results on elliptic curves, which are better understood.
122
123\subsection{Elliptic Curve Definitions}
124
125An elliptic curve $E$ over a field $K$ is the locus of points $(x,y)$ satisfying the \emph{Weierstrass equation}
126$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,$$
127with coefficients $a_i\in K$. A notion of summation of points turns this set into an abelian group. We won't give the group law algorithm here explicitly, but refer the reader to \cite{Silv}, Ch. III \S 2. The motivation for the composition law is that Bezout's theorem indicates that a line intersects this degree three curve in exactly three points (counting multiplicity in the special case of tangency). Therefore there is a unique third point $R$ on both the curve and the line connecting any given points $P$ and $Q$. Also, given a base point $\scro$, there is a unique point $R'$ on the curve that is on the line connecting $\scro$ and $R$. Define $R'=P\oplus Q$. This binary operation gives $E$ the structure of an abelian group. (All the group laws are straightforward to observe except associativity, which one can demonstrate in a technical argument with the explicit equations for the curve and lines, but which we happily avoid here.)
128
129This characterization is already an abuse of notation, since technically the curve ought to be in projective space for the discussion above to make any sense at all. However, the affine curve has a unique, well-defined projective closure, so rigor is not lost. This aside is by way of excusing the fact that sometimes we'll be lax and refer to the point $(x,y)$ on a curve instead of the point $[X:Y:Z]$ on the projectivized curve.
130
131To further simplify things for us, if the field is not of characteristic 2 or 3, we can make a rather messy substitution
132\begin{align}\label{simple1}
133x&\longmapsto \frac{1}{36}(x-3a_1^2-12a_2)\\
134y&\longmapsto \frac{1}{216}(y-a_1x-a_3)\label{simple2}
135\end{align}
136to reduce the equation to the form $y^2=x^3+Ax+B$, where $A$ and $B$ hide the ugly expressions
137\begin{align*}
138A=&-27a_1^4-216a_1^2a_2-432a_2^2+648a_1a_3+1296a_4,\\
139B=&54a_1^6+648a_1^4a_2+2592a_1^2a_2^2-1944a_1^3a_3+3456a_2^3-3888a_1^2a_4-7776a_1a_2a_3\\
140&+11664a_3^2-15552a_2a_4+46656a_6.
141\end{align*}
142When we need an explicit Weierstrass equation, we'll often use the simplified version for convenience, keeping in mind that the arguments would need to be modified to fit the full six-coefficient Weierstrass form if we are going to be able to apply them to curves over fields with characteristic 2 or 3. In the simplified case, however, define the \emph{discriminant} of the Weierstrass equation to be $\Delta = -16(4A^3+27B^2)$. The general form of the discriminant is not enlightening for our purposes, but note that it is not just $\Delta$ with the reverse substitutions for $A$ and $B$ (because then it would always be a multiple of 16). The discriminant of an elliptic curve is 0 if and only if it is singular. The group law still applies to the non-singular points of a given singular curve, however.
143
144Finally, we remark that elliptic curves are abelian varieties of genus 1.
145
146\section{Formal Groups}
147
148\subsection{The Basics}
149
150The following machinery is given in Silverman \cite{Silv}, chapter IV. We try to be a little clearer in the explanations, sometimes filling in arguments that he elides. This is evident particularly when it comes to facts about the inverse map, which are assumed in the source without any mention of what exactly is being assumed. But they aren't, in fact, terribly tedious or difficult to prove, and so we prove them here.
151
152\begin{definition}\label{formal group}
153A \emph{formal group} $\scrf$ over a ring $R$ is a two-variable power series $F(X,Y)\in R[[X,Y]]$ satisfying the following five properties:
154\begin{enumerate}
155\item
156$F(X,Y)=X+Y+$ [terms of higher degree]. \label{degree}
157\item
158$F(X,F(Y,Z))=F(F(X,Y),Z)$.
159\item
160$F(X,Y)=F(Y,X)$.
161\item
162$F(T,i(T))=0$ has a unique solution for $i(T)\in R[[T]]$.
163\item
164$F(X,0)=X$, $F(0,Y)=Y$.\label{identity element}
165\end{enumerate}
166Observe that these five properties provide an abelian group structure without any explicit elements. (The second, third, and fourth properties are associativity, commutativity, and inverses, respectively, and the fifth provides an identity.) Instead of elements, there is only the power series $F$ which provides the recipe for the group operation: hence the name formal group."
167\end{definition}
168
169\begin{definition}
170A \emph{homomorphism} between two formal groups $f:\scrf\to\scrg$ defined over a ring $R$ is a one-variable power series $f(T) \in R[[T]]$ with no constant term such that $f(F(X,Y))=G(f(X),f(Y))$. The formal groups $\scrf$ and $\scrg$ are \emph{isomorphic} if there exists a homomorphism $f$ as above and a reverse homomorphism $g$ such that $f(g(T))=g(f(T))=T$.
171\end{definition}
172
173For example, define the map $[0]:\scrf\to\scrf$ by $[0](T)=0$. It is a homomorphism because
174$$[0](F(X,Y))=0=F(0,0)=F([0](X),[0](Y)).$$
175That is an entirely trivial example, but it is important: we can use it to define a map $[m]:\scrf\to\scrf$ inductively for any integer $m$ by
176\begin{align*}
177[m](T)&=F([m-1](T),T) &(m>0)\\
178[m](T)&=F([m+1](T),i(T)) &(m<0)
179\end{align*}
180We'll show briefly that these maps, which are (perhaps confusingly at first) called \emph{multiplication by m}, are all homomorphisms (we'll say that the case $m>0$ will do to show this; the other case is similar).
181\begin{proof}
182Assume that $[m-1]$ is known to be a homomorphism. Then
183\begin{align*}
184[m](F(X,Y))&=F([m-1](F(X,Y)),F(X,Y))\\
185&=F(F([m-1]X,[m-1]Y),F(X,Y)) \\
186&=F([m-1]X,F([m-1]Y,F(X,Y))) \\
187&=F([m-1]X,F(F([m-1]Y,Y),X)) \\
188&=F([m-1]X,F([m]Y,X)) \\
189&=F(F([m-1]X,X),[m]Y)\\
190&=F([m]X,[m]Y)
191\end{align*}
192where we've used the commutativity and associativity properties of the formal group a few times. In particular, $[m]$ is a homomorphism as we'd wanted.
193
194We should also note that in the downward induction we need to use the fact that $i:\scrf\to\scrf$, the unique inverse map associated to the formal group $\scrf$, is also a homomorphism. To see this, consider the expression
195$$F(F(X,iX),F(Y,iY))=F(F(X,Y),F(iX,iY))$$
196obtained by repeated application of commutativity and associativity of the formal power series. The left hand side is trivially 0. On the right, remember that $iF(X,Y)$ is the unique power series for which $F(F(X,Y),iF(X,Y))=0$. So $iF(X,Y)=F(iX,iY)$ and $i:\scrf\to\scrf$ is a homomorphism. The second induction therefore follows exactly as the first did.
197\end{proof}
198
199We will now prove two facts about the multiplication-by-$m$ map.
200
201\begin{proposition}\label{mult}
202For any integer $m$, the map $[m](T)= mT +$ [higher order terms].
203\end{proposition}
204
205\begin{proof}
206We first induce to obtain the result for non-negative $m$. It is clear that $[0](T)=0$ works. Now assume the result about the map $[m-1]$. Using the recursive definition and then property \ref{degree} of formal groups, we have
207\begin{align*}
208[m](T)&=F([m-1](T),T)\\
209&=[m-1](T)+T+\textrm{[higher order terms]}\\
210&=(m-1)T+T \mod T^2
211\end{align*}
212which is exactly what we needed.
213
214For the downward induction, we'll need to establish a minor result about $i(T)$ again, namely that it has no constant term. This follows from property \ref{identity element} of formal power series:
215$$i(0)=F(0,i(0))=F(T,i(T))|_{T=0}=0.$$
216
217Now, notice that
218$$0=F(T,i(T))=T+i(T)+\ldots$$
219where the [$\ldots$] are higher order combinations of $T$ and $i(T)$. Rearrange to find
220$$i(T)= -T+\ldots$$
221We showed that $i(T)$ has no constant term, so everything on the right hand side after the $-T$ vanishes modulo $T^2$. Now the downward induction is clear.
222\end{proof}
223
224For the second proposition, we have to first prove a standard result about formal power series.
225
226\begin{lemma}\label{powerinverse}
227Let $R$ be a ring and $f(T) \in R[[T]]$ a one-variable power series, with $f(T)=aT +$ [higher order terms]. If $a\in R^*$, then there is a unique power series $g(T)\in R[[T]]$ that composes with $f(T)$ so that $f(g(T))=T$. This power series happens to also satisfy $g(f(T))=T$.
228\end{lemma}
229
230\begin{proof}
231The goal is to produce a power series that satisfies $f(g(T))\equiv T$ modulo $T^n$ for any $n>1$. We'll do this by explicitly constructing the polynomials $g_n(T)$ which we define to be $g(T)$ modulo $T^{n+1}$; since they'll be inductively well-defined, that will give us a precise definition of the formal power series $g(T)$.
232
233First, take $g_1(T) = a^{-1}T$. Clearly $f(g_1(T))=a(a^{-1}T)+\ldots\equiv T$ mod $T^2$, as we'd wanted.
234
235Next, assume $f(g_{k-1}(T))\equiv T$ mod $T^{k}$. We'll have $g_k = g_{k-1}+ rT^k$ for some $r\in R$. Observe that
236\begin{align*}
237f(g_k(T))&=f(g_{k-1}(T)+rT^k)\\
238&\equiv f(g_{k-1}(T)) + arT^k \mod T^{k+1}
239\end{align*}
240since all cross-terms contributed by the $rT^k$ monomial have degree $>k$. Remember by our initial assumption that
241\begin{align*}
242f(g_{k-1}(T))&\equiv T\mod T^k\\
243&\equiv T + bT^k \mod T^{k+1}
244\end{align*}
245for some element $b\in R$. Then put it all together to get
246$$f(g_k(T)) \equiv T + bT^k + arT^k \mod T^{k+1}.$$
247So there's a unique choice $r = -b/a$ at each stage, well-defined because $a$ is a unit, such that $f(g_k(T)) \equiv T \mod T^{k+1}$, and the formal power series $g(T)$ therefore exists.
248
249Now that we know $f(g(T))=T$, we also have $g(f(g(T))=g(T)$, which is a formal identity in $R[[g(T)]]$. In particular, under the formal map of variables $g(T)\mapsto T$, we find that $g(f(T))=T$ as we'd claimed.
250
251Finally, to show uniqueness of $g$, assume there's another power series $h(T)$ such that $(f\circ h)(T)=T$; then
252$$g(T)= g \circ (f\circ h)(T)=(g \circ f)\circ h(T)=h(T).$$
253\end{proof}
254
255Now we can prove the second fact, which quite naturally follows from the above.
256
257\begin{proposition}\label{multisisom}
258If the integer $m$ is a unit in $R^*$, then $[m]:\scrf\to\scrf$ is an isomorphism.
259\end{proposition}
260
261\begin{proof}
262>From Proposition \ref{mult}, $[m](T)=mT+\ldots$; knowing that $m\in R^*$ and applying Lemma \ref{powerinverse}, there is a power series $[m]^{-1}(T)$ such that $[m]^{-1}([m](T))=[m]([m]^{-1}(T))=T$. The only thing remaining is to show that $[m]^{-1}$ is a homomorphism $\scrf\to\scrf$. We know $[m]$ is a homomorphism, so
263\begin{align*}
264[m]^{-1}F(X,Y)&=[m]^{-1}F([m][m]^{-1}(X),[m][m]^{-1}(Y))\\
265&=[m]^{-1}[m]F([m]^{-1}(X),[m]^{-1}(Y))\\
266&=F([m]^{-1}(X),[m]^{-1}(Y))
267\end{align*}
268gives us what we need.
269\end{proof}
270
271\subsection{So where are the elements?}
272
273The natural thing to do after defining a formal group is to think about supplying elements that can follow this recipe for their group structure.
274
275Recall (Atiyah-Macdonald \cite{Atiyah}, ch. 10) that a complete local ring $R$ with maximal ideal $M$ is defined as a ring that is equal to its natural inverse limit:
276$$R \isom \lim_{\longleftarrow} R/M^n.$$
277If we have a power series $f$ and an element $\alpha \in M$, we can use these to produce a unique element $\alpha_n\in M^n$, since truncating the power series at the degree $n$ term is fine (all higher terms will be congruent to zero modulo $M^n$). The completeness of the ring indicates that this sequence of elements $(a_1,\ldots,a_n,\ldots)$ is under isomorphism an element of $R$. In particular, this argument extends to show that with $R,M$ as above, a power series in two variables $F(X,Y)$ given as the group law for a formal group will converge for $X,Y \in M$. The set $M$ is thus made into a group, denoted $\scrf(M)$, with the group composition $x\oplus y = F(x,y)$ and inverse $\ominus x = i(x)$. The closure of the group is shown by property \ref{degree} of formal power series (Definition \ref{formal group}): the equation tells us that $x\oplus y \equiv 0$ modulo $M$. Finally, $\scrf(M^n)$ is a subgroup of $\scrf(M)$, since it is closed under composition by precisely the same argument.
278
279\begin{remark}
280Now we see exactly why the multiplication-by-$m$ map was so called; the map on the formal group induces a map on the associated group $\scrf(M)$ which acts by composing an element with itself $m$ times.
281\end{remark}
282
283We now prove a major result about torsion in groups associated to formal groups.
284
285\begin{proposition}\label{noems}
286Let $\scrf$ be a formal group over a complete local ring $R$ with maximal ideal $M$, and let $k$ be the residue field $R/M$. Let the characteristic of $k$ be $p$. (Zero characteristic is allowed.) Then every element of finite order in $\scrf(M)$ has order a power of $p$.
287\end{proposition}
288
289\begin{proof}
290For positive $p$, if there were a point $r\in M$ that served as a counterexample, it would have order $p^k m$ for some $m$ relatively prime to $p$. There would then also be a point of order $m$, namely $[p^k] r$. Therefore it is enough to prove that there are no points of order $m$ with $p$ and $m$ relatively prime. (For $p=0$, we have to prove that there are no points of order $m>0$ at all.)
291
292Then let $m$ be a positive integer chosen as above, and let $x$ be a point of order $m$, so $[m](x)=0$. Observe that $m$ cannot be an element of the ideal $M$, because then $R/M$ would have characteristic dividing $m$, which is not permitted. Therefore $m\in R^*$, and applying Proposition \ref{multisisom}, the map $[m]:\scrf\to\scrf$ is an isomorphism. That's all we need, because it induces an isomorphism of groups $[m]:\scrf(M)\to\scrf(M)$, whose kernel consists only of the point $0$, so $x$ must be trivial.
293\end{proof}
294
295\subsection{More Theory of Formal Groups}
296
297If the ring is equipped with a discrete valuation $v$, we can say something further about the torsion in the group $F(M)$.
298
299\begin{proposition}\label{valp}
300Let $R$ be a complete local ring with maximal ideal $M$, and with a discrete valuation $v$. Let $p$ be the characteristic of the field $R/M$. Let $\scrf$ be a formal group over $R$, with associated group $\scrf(M)$. Given an element $x\in\scrf(M)$ of prime power order $p^n$, we have
301$$v(x) \le \frac{v(p)}{p^n-p^{n-1}}.$$
302\end{proposition}
303
304This proposition relies on the notion of invariant differentials for formal groups, the introduction of which may take us a bit too far afield. It is proved quite concisely in \cite{Silv}, ch. IV, \S4 and \S6.
305
306\section{Application to Elliptic Curves: the Finitude of $E(K)\tor$}
307
308In this section we will demonstrate that torsion subgroups of elliptic curves over number fields are finite, as a consequence of a demonstrably injective map into a finite group. This result also follows from the Mordell-Weil theorem (which states that the group $E(K)$ of an elliptic curve over a number field is finitely generated). However, the argument that we will present here is independent and easier to prove. Additionally, it gives an effective bound (since one can compute the number of points in the reduction of the curve easily), where the Mordell-Weil theorem does not.
309
310\subsection{An Elliptic Curve's Formal Group}
311
312In what follows, we will use the simplified form of the elliptic curve, so the full generality of characteristic 2 and 3 is denied us; however, \cite{Silv}, IV \S 1, does that rather well. In choosing to work with the simplified curve, we hope to trade full generality for some added intuition, as the algebraic contortions with the five-parameter form are not particularly enlightening. The following section is mostly sketches to get a grasp of what's going on; the formulas themselves will be elided most of the time.
313
314Given an elliptic curve $E:y^2=x^3+Ax+B$, perform the change of coordinates $s=\frac{1}{y}$, $t=\frac{x}{y}$. The additive identity becomes $(0,0)$ under the new $(s,t)$-coordinate system. The equation of the curve becomes
315$$s=t^3+Ats^2+Bs^3.$$
316What if we want $s$ as a function entirely of $t$? The natural thing to do is recursively substitute the equation for $s$ back into itself, as follows:
317\begin{align*}
318s&=t^3+Ats^2+Bs^3\\
319&=t^3+At(t^3+Ats^2+Bs^3)^2+B(t^3+Ats^2+Bs^3)^3\\
320&=t^3 + At^7+ Bt^9 + s^2(2A^2t^5+3ABt^7) + s^3(2ABt^4+3B^2t^6)\\
321&\ \ +s^4(A^3t^3+3A^2Bt^5)+s^5(2A^2Bt^2+6AB^2t^4)\\
322&\ \ +s^6(AB^2t+3B^3t^3+A^3Bt^3)+s^7(A^2B^2t^2)+s^8(3AB^3t)+s^9(B^4).
323\end{align*}
324This repeated substitution gets us a polynomial in $t$ followed by a polynomial in $s$ and $t$; by continuing inductively, the powers of $s$ get larger and larger, and thus contribute larger and larger powers of $t$, so the procedure feels like it should converge to an element of the ring of formal power series $R[[t]]$. In fact, it does; this is a consequence of Hensel's lemma (see Silverman \cite{Silv}, IV Lemma 1.2.).
325
326In particular, there is a unique formal power series $s(t)\in R[[t]]$ such that $s(t)=t^3+ats(t)^2+Bs(t)^3$. Letting the expression $f(s,t)=t^3+Ats^2+Bs^3$, then $s(t)$ satisfies $f(s(t),t)=s(t)$. We know the first few terms as before, after a few self-substitutions:
327\begin{align}
328s(t)&=t^3 + At^7 + Bt^9 + 2A^2t^{11} + 5ABt^{13}\label{st}\\
329&\ \ + (5A^3 + 3B^2)t^{15} + 21A^2Bt^{17}+\ldots \notag
330\end{align}
331Now substituting back for the standard coordinates, we have
332\begin{align*}
333x(t)&=\frac{t}{s(t)}=\frac{1}{t^2} - At^2 - Bt^4 - A^2t^6 - 3ABt^8 + (-2A^3 - 2B^2)t^{10} - 10A^2Bt^{12} \ldots\\
334y(t)&=\frac{1}{s(t)}=\frac{1}{t^3} - At - Bt^3 - A^2t^5 - 3ABt^7 + (-2A^3 - 2B^2)t^9 - 10A^2Bt^{11} \ldots
335\end{align*}
336
337Because of our discussion above on the convergence of formal power series, if $K$ is a complete local field with $R$ and $M$ as we've been using them above, then taking $t$ to be an element of $M$ gets us a point in $E(K)$. (The fact that these are Laurent series doesn't change anything about the convergence because we're only adding a finite number of terms to a series known to be convergent.) Furthermore, the map is injective, since $t=x/y$ is the inverse. We can think of $t$ as a parameter for all points on the curve with $t=x/y\in M$.
338
339Finally, since there is a group law for the explicit addition of points $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2) \in E$, we can make a corresponding formula for the formal addition of indeterminates $t_1$ and $t_2$, which we eschew here, since we have made it a point to avoid the group law calculations above. However, the addition formula is a formal power series in two variables $F(t_1,t_2)$. It is constructed to obey all the properties that the usual addition on elliptic curves obeys; therefore it is the basis for a formal group.
340
341\begin{definition}
342The formal group $\hat{E}$ (over $R$) associated to the elliptic curve $E$ (over $K$) is the formal power series $F$ defined above, which gives the group law.
343\end{definition}
344
345\subsection{Reduction on Elliptic Curves}
346
347Let $K$ be a local field with discrete valuation $v$, and $R$ be the ring of integers of $K$. Recall that since $R = \{x\in K: v(x)\ge0\}$, we can decompose $R$ into a group of units $R^*$ with valuation 0 and the set of elements with positive valuation; these last constitute an ideal in $R$, which is clearly the unique maximal ideal since it contains all non-units, so we'll call it $M$ as before. (The ring of integers of a local field is thus a local ring.)
348
349Now let $M$ be $\pi R$ for some uniformizing element $\pi$, and for convention's sake let's say that the discrete valuation has been normalized so that $v(\pi)=1$. Finally, let $k$ be the field $R/M$. In a bit we'll be able to think about moving a curve's base field from $K$ to $k$. But first we have to put the curve in canonical form.
350
351\begin{definition}
352Given an elliptic curve $E/K$ with Weierstrass equation $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$, we say the equation is \emph{minimal} if $a_i \in R$ and $v(\Delta)$ is minimized over all possible changes of coordinates.
353\end{definition}
354
355\begin{remark}\label{minimal}
356There is a minimal equation representing any curve, because the valuation $v$ is discrete.
357\end{remark}
358
359\begin{remark}
360The explicit change of coordinates allowed is given in \cite{Silv}, ch. III, \S 1. Observe that we can also minimize a simplified Weierstrass equation $y^2=x^3+Ax+B$ if we're not working over a local field with residue characteristic 2 or 3: certainly we can minimize the valuation of the determinant in the five-parameter form, then do the transformations \ref{simple1} and \ref{simple2} in order to put it in simplified form. The determinant changes by a factor of $1/6^{12}$, so it remains minimal with our valuation. Therefore in what follows we'll consider elliptic curves of the form $y^2=x^3+Ax+B$, with $A,B\in\Z$.
361\end{remark}
362
363\begin{definition}
364The natural \emph{reduction map} is the map $R\to R/M$, which action on the elements of $R$ we denote by $r\mapsto \tilde{r}$. The reduction of an elliptic curve over a local field $K$ is the curve produced by the action of the reduction map on a minimal Weierstrass equation for $E$:
365$$[E:y^2=x^3+Ax+B] \longmapsto [\tilde{E}:y^2=x^3+\tilde{A}x+\tilde{B}].$$
366The reduction on a projective point $P=[x:y:z]$ is the reduction map on the coordinates of the point once it has been put into canonical form, that is, with all coordinates in the ring of integers $R$ and minimized so that one or more is a unit. This is required for the image of the point to be well-defined. Then $\tilde{P}$ is a point on $\tilde{E}/k$.
367\end{definition}
368
369Since the curve's equation was chosen to be minimal, the map between curves is well-defined, up to a change of coordinates over the residue field $k$. It is a homomorphism because it maps lines in $\p^2(K)$ to lines in $\p^2(k)$, and therefore the group laws are preserved.
370
371The reduced curve is not guaranteed to be non-singular. If the discriminant of a Weierstrass equation is 0, then its elliptic curve has singular points; however, it is a fact that the remaining non-singular points still form an abelian group. Let $\tilde{E}\ns(k)$ be the group of non-singular points of $\tilde{E}$. Then its inverse image in the set $E(K)$ is denoted $E_0(K)$, the points of non-singular reduction. Also define $E_1(K)$ to be the kernel of reduction, that is, the points that map to the identity on the reduced curve.
372
373\begin{proposition}\label{exact}
374The sequence
375$$0\to E_1(K)\to E_0(K) \to \tilde{E}\ns(K),$$
376where the second map is inclusion and the last map is reduction, is exact.
377\end{proposition}
378
379\begin{proof}
380This follows from the definitions. Since the map $E_1(K)\to E_0(K)$ is inclusion, as observed above, it is injective. Then the kernel of the reduction map is exactly $E_1(K)$ by definition.
381\end{proof}
382
383\begin{remark}
384In fact, the reduction map is surjective (see \cite{Silv}, ch. VII, proposition 2.1---the proof uses Hensel's lemma for the lifting), so we can extend the above into a short exact sequence of groups, but the surjectivity is not essential to what follows.
385\end{remark}
386
387Finally, we put together the reduction map and the formal group associated to the curve.
388
389\begin{proposition}\label{isom}
390Let $\hat{E}$ over $R$ be the formal group associated to $E$ over $K$, and let $s(t)$ be the power series as in equation \ref{st}. Then the map $\hat{E}(M) \to E_1(K)$ given by $t \longmapsto (\frac{t}{s(t)},\frac{1}{s(t)})$ is an isomorphism. In this map, whose image is not explicitly composed of projective points, we assert that the point $z=0$ is mapped to the additive identity $(0,\infty)$.
391\end{proposition}
392
393\begin{proof}
394We already know that the image points satisfy the Weierstrass equation, and we have remarked that $t\in M$ makes the defining series converge to a point in $E(K)$ (it is in $E(K)$ and not $E(M)$ because the series is Laurent). To see that it is actually in $E_1(K)$, notice that after clearing denominators of elements of $M$ in the point $[\frac{t}{s(t)}:\frac{1}{s(t)}:1]$, the first and third coordinates have positive valuation while the second is a unit, and therefore the point reduces to $[0:1:0]$ on the curve $\tilde{E}(k)$. The explanation in \cite{Silv} (ch. VII, \S 2) of this fact is not particularly clear.
395
396It remains only to produce an inverse. We mentioned a putative one in our original remarks, namely $t=x/y$, so it suffices to show that given $(x,y)\in E_1(K)$, $x/y\in M$. But this is evident by inspection: if $[x:y:1]$ is to reduce to $[0:1:0]$ as above, we must have $x$ and $y$ with negative valuation, and, further, $v(y)<v(x)$. Therefore $v(x/y)>0$ and so $x/y\in M$.
397\end{proof}
398
399Now we have the material needed to prove the main result.
400
401\begin{theorem}\label{inj}
402Let $E$ be an elliptic curve over $\Q$, and $p>3$ a prime. Assume that $p$ does not divide the discriminant $\Delta$, so the reduced curve $\tilde{E}$ is non-singular. Then the induced map on torsion $E(\Q)\tor \to \tilde{E}(\F_p)\tor$ is injective.
403\end{theorem}
404
405The theorem is also true for $p=3$, but false for $p=2$. We will discuss this further below. This proof works verbatim for $p=3$.
406
407\begin{proof}
408Consider the extension our curve $E(\Q)\subset E(\Q_p)$ to a local field in which it is embedded. We can now apply all the theory we've been assembling, with $K=\Q_p$. We have the standard $p$-adic valuation, and the maximal ideal is $M=(p)$.
409
410First, with the aid of Proposition \ref{valp}, we show that there cannot be any elements of order $p$ in the formal group $\hat{E}(M)$. Assume an element $x\in M$ has order $p$. Then by Proposition \ref{valp}, we have $0 < v(x) \le \frac{v(p)}{p-1}$. Since $v$ is the standard $p$-adic valuation on the rational numbers, $v(p)=1$ and $v(x)$ is an integer. Since $p>2$, we have $0< v(x) \le \frac{1}{p-1} < 1$, a contradiction.
411
412Next, recall that there cannot be any elements of order relatively prime to $p$ in $\hat{E}(M)$, by Proposition \ref{noems}. Hence we conclude that there are no torsion points of any kind in the group $\hat{E}(M)$.
413
414By Proposition \ref{isom}, $\hat{E}(M)\isom E_1(\Q_p)$ and thus there is no torsion in the group $E_1(\Q_p)$.
415
416Finally, we avail ourselves of the exact sequence of Proposition \ref{exact}, which tells us that the group $E_1(\Q_p)$ is injectively mapped into $E_0(\Q_p)$, thus has a torsion-free image under this map, and this image is exactly the kernel of the reduction map into $\tilde{E}\ns(\F_p)$. Since the reduced curve is non-singular, $\tilde{E}(\F_p)=\tilde{E}\ns(\F_p)$ and $E_0(\Q_p)=E(\Q_p)$. Thus we have an exact sequence
417$$0\to \hat{E}(M) \to E(\Q_p) \to \tilde{E}(\F_p).$$
418In particular, all the points in $E(\Q_p)$ that reduce to the identity in $E(\F_p)$ are non-torsion. So the torsion in $E(\Q_p)$ and hence in $E(\Q)$ must be mapped injectively into $\tilde{E}(\F_p)$.
419
420\end{proof}
421
422\begin{corollary}
423The order of a torsion subgroup of an elliptic curve over $\Q$ is finite.
424\end{corollary}
425
426\begin{proof}
427There exist infinitely many primes of good reduction, since any prime that does not divide the discriminant is a prime of good reduction. Choose $p>3$ and reduce. The reduction map is injective on torsion.
428\end{proof}
429
430\begin{remark}
431A similar argument works on elliptic curves over finite extensions of $\Q$. It may become necessary to choose a higher prime so that $\frac{v(p)}{p-1}<1$, since the valuation of $p$ may not be 1 in the corresponding extension of $\Q_p$, but once a prime has been found that works, the torsion subgroup is automatically finite.
432\end{remark}
433
434We should observe that if we assume the preliminary results for the five-parameter Weierstrass equations (not a big leap), Theorem \ref{inj} will also hold for $p=3$ but not $p=2$, since $v(2)=1$ does not satisfy the condition $v(p)<p-1$.
435
436The elliptic curve $E:y^2+xy+y=x^3+x^2-10x-10$ is an example of a curve with good reduction at $p=2$ (since $\Delta=3^45^4$) for which the corresponding reduction map is not injective on torsion. Checking the torsion by computer, we find that it is equal to $\Z/8\Z$; however, there are only 5 points that could possibly be in $E(\F_2)$. This is not a contradiction, however, because of the above observation.
437
438To be explicit, we check the torsion over $\F_2$ as well, using the computer algebra package MAGMA:
439\begin{verbatim}
440> E := EllipticCurve([ 1, 1, 1, -10, -10 ]);
441> F := ChangeRing(E,GF(2));
442> AbelianGroup(F);
443Abelian Group isomorphic to Z/4
444Defined on 1 generator
445Relations:
446     4*$.1 = 0 447\end{verbatim} 448 449A brief aside: how did we know that there could be at maximum five points on$E(\F_2)$? It's not hard for$\F_2$because there are four points in$(\F_2)^2$together with the point at infinity. But for larger primes, we can use the Hasse bound, proved in the 1930s; we state the result in a slightly weakened form here. The full result includes all finite fields. 450 451\begin{theorem}[Hasse] 452$\#E(\F_p)\le p+2\sqrt{p}+1.$453\end{theorem} 454 455Now apply this to the prime$p=3$: it says that there at most seven points on an elliptic curve over$\F_3$. However, we've seen more torsion than that in an elliptic curve over$\Q$; just take the example above. However, the reason we can't apply the theorem is that the curve doesn't have good reduction at$p=3$. This gives us an \emph{a priori} bound on torsion. If the discriminant is not divisible by 3, then the torsion subgroup has seven points or fewer. 456 457\section{Statements of Two Theorems About Torsion} 458 459In 1977-8, B. Mazur published a paper \cite{Mazur} that completely answered the question of what torsion subgroups are possible for an elliptic curve over$\Q$. We cite his result here. 460 461\begin{theorem}[Mazur]\label{Mazur} 462Let$E/\Q$be an elliptic curve. Its torsion subgroup$E(\Q)\tor$must be one of the following fifteen groups:$\Z/n\Z$for$n \in \{1,2,3,4,5,6,7,8,9,10,12\}$,$\Z/2\Z \times \Z/n\Z$for$n \in \{2,4,6,8\}$. Further, all of these groups are represented as the torsion groups of some elliptic curve over$\Q$. 463\end{theorem} 464 465In 1996, L. Merel proved a theorem that resolved a long-standing question about the torsion groups of elliptic curves over number fields$K$. The proof of it is also well beyond the scope of this paper (see \cite{Merel}). 466 467\begin{theorem}[Merel]\label{Merel} 468Let$E/K$be an elliptic curve, where the number field$K$satisfies$[K:\Q]=d>1$. Then any torsion point has order$n<d^{3d^2}$. 469\end{theorem} 470 471In particular, Merel uses this theorem along with previous work of Faltings and Frey to demonstrate that there is a finite number of possible torsion groups of elliptic curves over a degree$d$extension of$\Q$. This result is stunningly general. However, although one knows that the list is finite, and for a given$d$one can take many elliptic curves, compute their torsion groups, and compile a list of the torsion that has appeared, it is not known how to prove that any such list would actually be complete. Mazur's method does not work on number fields. 472 473\section{Torsion Points on Elliptic Curves: Algorithmic Questions} 474 475 476\subsection{The Nagell-Lutz Theorem} 477 478Now since$E(K)\tor$is finite, the natural follow-up question is: is it easily computable? For elliptic curves, the answer is well known. Two papers published in the 1930s, one by E. Lutz and the other by T. Nagell, give a clear idea of what torsion points are possible. Their conclusion is presented in the following theorem: 479 480\begin{theorem} 481Let$E$be an elliptic curve over$\Q$with Weierstrass equation$y^2 = x^3 + Ax+B$, with$A$,$B$integers. Then the coordinates of a non-zero torsion point$P=(x_0,y_0)\in E(\Q)$are in$\Z$. Furthermore,$P$is either of order$2$, or else$y_0^2$divides$4A^3+27B^2$. 482\end{theorem} 483 484This characterization of torsion points on elliptic curves over$\Q$limits the search space. Practically, points of order 2 are easy to find, because in that case$y_0=0$; and the second condition ensures that if$P$is of higher order then there are only finitely many possibilities to check for the value of$y_0$. Therefore there are finitely many pairs$(x_0,y_0)$to check, since each choice of$y_0$produces a monic integral cubic equation for$x_0$, any integral solution of which will divide the constant term,$B-y_0^2$. We should remark that this is yet another proof of finiteness of the torsion subgroup, independent of either of the two methods previously mentioned. 485 486The first step is to show that the coordinates of a torsion point are integers. 487 488\begin{proof} 489Let$m$be the smallest integer for which$mP=\scro$. If$m=2$, then$P = -P =(x_0,-y_0)$. So$y_0=0$and hence$x_0\in \Z$, since it is the root of a monic polynomial with integral coefficients. 490 491Now assume$m>2$. Our goal will be to show that for every embedding$E(\Q)\hookrightarrow E(\Q_p)$, the coordinates of our point have nonnegative valuation in the local field. Let$M$be the maximal ideal$(p)$to avoid notation confusion. 492 493First, if$v(x_0)\ge 0$, then the Weierstrass equation tells us that$v(y_0)\ge 0$as well. So we'll assume$v(x_0)<0$and find a contradiction. 494 495We observe that the curve has a minimal model, by Remark \ref{minimal}, so the reduction map is well defined. By applying the valuation to both sides of the Weierstrass equation again,$v(y_0)<v(x_0)<0$, and so the projective point$[x_0:y_0:1]$reduces to the point$[0:1:0]$under the map$E(\Q_p)\to \tilde{E}(\F_p)$(we've used this reasoning before, in the proof of Proposition \ref{isom}). So$(x_0,y_0)$is a member of$E_1(\Q_p)\isom \hat{E}(M)$. 496 497We can now apply our usual propositions about torsion in formal groups. Proposition \ref{noems} tells us that if$m$is relatively prime to$p$we already have our contradiction (there can't be such a torsion point), and Proposition \ref{valp} tells us that if$m=p^n$is a power of$p$, then$0 < v(x/y) \le \frac{1}{p^n-p^{n-1}}$. But for$m>2$this right hand side is less than 1, so we have our contradiction. 498 499Because the coordinates, when thought of as elements of$\Q_p$, are in the ring of integers$\Z_p$for every$p$, they must be in$\Z\subset \Q$. 500 501\end{proof} 502 503It remains only to verify that$y_0^2$divides$4A^3+27B^2$when the torsion point is not of order 2. This is essentially a computation using the duplication formula for a point under the formal addition law on the elliptic curve, following \cite{Silv}. 504 505\begin{proof} 506Assume$2P\ne \scro$. Write$2P= (x_1,y_1)$. Since it is also a torsion point,$x_1,y_1\in\Z$. We can use the duplication formula for a point to express$x_1$in terms of$x_0$: 507$$x_1=\frac{x_0^4-2Ax_0^2-8Bx_0+A^2}{4(x_0^3+Ax_0+B)}.$$ 508Notice that the denominator is just$y_0^2. Now we produce a polynomial identity by using the Euclidean algorithm on the numerator and denominator of the fraction above. Since the fraction is reduced, we'll get some linear combination of the two polynomials equal to a constant term. 509\begin{align*} 510(3X^2+4A)(X^4-2AX^2-8BX+A^2)&-\\ 511(3X^3-5AX-27B)(X^3+AX+B)&=4A^3+27B^2. 512\end{align*} 513Observe that the constant term works out to exactly the value we had above. So all we have to do is plugX=x(P)$into this polynomial identity, and simplify using the relations for$x_1$and$y_0^2: 514\begin{align*} 515(3x_0^2+4A)(x_1\cdot 4y_0^2)&-\\ 516(3x_0^3-5Ax_0-27B)(y_0^2)&=4A^3+27B^2. 517\end{align*} 518All of the variables in this equation are integers. Now it is clear thaty_0^2$divides the left hand side; therefore it must divide the right. 519\end{proof} 520 521\subsection{Doud's Algorithm} 522 523The naive approach to finding the torsion on elliptic curves over$\Q$, then, would be to check each of the finitely many pairs of coordinates$(x,y)$suggested by the Nagell-Lutz theorem, knowing by Mazur's Theorem (\ref{Mazur}) that a torsion point cannot have order greater than 12. In fact, this approach was widely implemented until fairly recently. J.W. Cremona's 1997 book \cite{Cremona} of elliptic curve algorithms gave it as the standard recipe for the computation of torsion. However, Darrin Doud \cite{Doud} points out two disadvantages to this method: first, it requires factorization of the discriminant, which may be large; second, for each square divisor of the discriminant, a cubic equation must be solved, and there may be many such divisors. (Remember that the number of divisors grows exponentially in the number of prime factors of the discriminant.) 524 525In its place, Doud suggests using the Weierstrass$\wp$function associated to a complex lattice$\Lambda\in\C$. Given the lattice, we associate to it the doubly periodic function 526$$\wp(z) = \frac{1}{z^2} + \sum_{\alpha\in\Lambda,\alpha\ne0}\left(\frac{1}{(z-\alpha)^2}-\frac{1}{\alpha^2}\right).$$ 527It is a standard result (see \cite{Silv}, chapter VI, \S3-5) that for the curve associated to the lattice, there is an isomorphism of groups$\phi: \C/\Lambda\to E(\C)$given by$\phi(z)=[\wp(z),\wp'(z),1]$. (Here the association of curve and lattice means that the curve$E$has Weierstrass coefficients that come from the Laurent series expansion of the function$\wp$, which depends on$\Lambda$. The precise algebraic way that the coefficients derive from the Laurent expansion is not essential to these remarks.) It is also known that the reverse association can be made: given an elliptic curve$E(\C)$, there is a lattice$\Lambda$such that the Laurent expansion of$\wp(z)$produces the Weierstrass coefficients for$E$. 528 529This suggests a natural way of checking for rational torsion in$E$, namely, consider the torsion points in$\C/\Lambda$and check to see whether their images are rational under the map$\phi$. Since the$n$-torsion points of$\C/\Lambda$are just$\frac{1}{n}\Lambda/\Lambda$, there isn't much to compute. The essential steps are therefore producing the lattice from the curve in the first place, which is quick relative to the rest of the algorithm, and then following the$n$-torsion from the lattice into the curve, checking to see whether they're integral points. Doud uses reduction (to determine a multiple of the torsion subgroup), in conjunction with Mazur's Theorem, to determine which values of$n$need to be observed. 530 531So Doud's method is rather quick. The only possible implementational pitfall to the algorithm is its analyticity; one needs to take care that one uses enough terms of the$\wp$function. Doud proved the required minimal precision for the algorithm in his paper, so the theory is fine. However, if the error bounds are not carefully implemented, one might falsely truncate the series for$\wp(z)$too early and produce a point on the curve that's close" to an integral point but not the one it's actually converging to. In that case the algorithm would return less torsion than actually present in the elliptic curve. In fact, this mistake was made in the original MAGMA implementation of Doud's method. In 2001, David Yeung observed that MAGMA's implementation of the algorithm was returning$\Z/8\Z$for the torsion subgroup of a curve he tested, while PARI produced$\Z/8\Z\times\Z/2\Z$. After a bit of confusion, the precision of the analytic method was increased, Doud's algorithm produced the right answer, and there have been no known further problems. The algorithm represents a significant speed increase over the Nagell-Lutz approach. 532 533\section{Background on Modular Abelian Varieties} 534 535\begin{definition} 536An \emph{algebraic group}$V$is an algebraic variety that is also a group, in that there are two regular maps, an inverse map$i:V\to V$and a composition map$m:V\times V \to V$, satisfying the usual group axioms on the points of$V$. An \emph{abelian variety}$A$is a projective, irreducible algebraic group. 537\end{definition} 538 539\begin{remark} 540In order to show that the algebraic variety with the above properties is abelian, one needs to use the fact that$A$is projective. The proof is found in Shafarevich \cite{Shaf}, 4.3. 541\end{remark} 542 543\subsection{Modular Groups \& Modularity} 544 545Good references for the following definitions include \cite{Stein}, \cite{Serre}, and \cite{Shimura}. 546 547Denote the open upper half of the complex plane by$\h \subset \C$. If we also include the points in$\Q$and a point$\{\infty\}$at infinity, we'll call it$\h^*$. (There's a slightly peculiar topology around the added points (known as cusps"), as there would have to be: see \cite{Shimura}, \S 1.3, for a reference on this, the Poincar\'{e} metric.) The group$\SL_2(\R)$acts on$\h^*$by linear fractional transformations, as follows: given an element$g=\smabcd{a}{b}{c}{d}$with$ad-bc=1$, and$z\in\h^*$, let 548$$gz=\frac{az+b}{cz+d}.$$ 549It is not too hard to check that$\h^*$is stable under the action of$\SL_2(\R)$, since 550$$\im(gz)=\im\left(\frac{(az+b)(c\bar{z}+d)}{|cz+d|^2}\right)=\frac{(ad-bc)\im(z)}{|cz+d|^2},$$ 551which is positive if the original imaginary part was positive. Notice that the element$\smabcd{-1}{\phantom{-}0}{\phantom{-}0}{-1}$is the only non-identity element that acts trivially on all of$\h^*$. So in what follows, we take care to mod out all the groups by$\pm 1$. Thus$\PSL_2(\R)$is a group that acts faithfully on$\h^*$. 552 553Now consider the discrete subgroup$\Gamma=\PSL_2(\Z)$of matrices with coefficients in$\Z$, again modulo$\pm1$. It is called \emph{the modular group}, and inherits an action on$\h^*$. For any integer$N$, we can produce three subgroups of the modular group$\Gamma$which also act on$\h^*: 554\begin{align*} 555\Gamma_0(N)&=\left\{\abcd{a}{b}{c}{d}\in\PSL_2(\Z)\st \abcd{a}{b}{c}{d}\equiv \abcd{*}{*}{0}{*}\mod N \right\}\\ 556\Gamma_1(N)&=\left\{\abcd{a}{b}{c}{d}\in\PSL_2(\Z)\st \abcd{a}{b}{c}{d}\equiv \abcd{1}{*}{0}{1}\mod N \right\}\\ 557\Gamma(N)&=\left\{\abcd{a}{b}{c}{d}\in\PSL_2(\Z)\st \abcd{a}{b}{c}{d}\equiv \abcd{1}{0}{0}{1}\mod N \right\} 558\end{align*} 559(Here the*$s can be anything.) Now if we quotient$\h^*$by the action of these groups, we get Riemann surfaces (\cite{Shimura}, \S 1.5). 560 561\begin{definition} 562For any positive integer$N$, the Riemann surfaces$X_0(N)$,$X_1(N)$, and$X(N)$are the spaces$\Gamma_0(N)\backslash \h^*$,$\Gamma_1(N)\backslash \h^*$,$\Gamma(N)\backslash \h^*$respectively. 563\end{definition} 564 565\begin{remark} 566The quotient is traditionally written on the left because the group action is written on the left. Also, because we have the inclusion of groups$\Gamma(N)\subset\Gamma_1(N)\subset\Gamma_0(N)$, the Riemann surfaces have corresponding relationships, in reverse order:$X(N)$is the biggest," and the other two are progressively quotients of it. Finally, these are algebraic curves over$\Q$(this is not obvious, and is proved in \cite{Shimura}, \S 6.7). For example, \cite{Steintalk} notes that$X_1(13)$is the desingularization of the projective closure of the affine curve$y^2=x^6+2x^5+x^4+2x^3+6x^2+4x+1$. 567\end{remark} 568 569Given a compact Riemann surface$X$of genus$g$, it has a Jacobian$J(X)$associated to it, which is a complex torus$\C^g/\Lambda$for a certain lattice$\Lambda$; we can also think about it as the space of divisors of degree zero on$X$up to linear equivalence. (Recall that a \emph{divisor} on$X$is a formal sum of points$p$in$X$with integer coefficients, 570$$D=\sum_{p\in X}n_pp,\ n\in\Z,$$ 571its \emph{degree} is the sum of those coefficients, 572$$\deg(D)=\sum_{p\in X}n_p,$$ 573and any meromorphic function$f: X\to\C$has a naturally associated divisor, namely 574$$(f)=\sum_{p\in X}(\ord_p(f))p.$$ 575Then$J(X)$is the space of divisors modulo the divisors of meromorphic functions.) It is not terribly obvious that the space of divisor classes of degree 0 has the structure of a complex torus; the result is due to Abel and Jacobi. For details of the isomorphism at a more or less elementary level, including what the lattice$\Lambda$is explicitly, see the author's junior paper \cite{Kleinerman}. 576 577We have defined three compact Riemann surfaces above, and briefly touched on the notion of the Jacobian. The next definition should therefore be intuitive. 578 579\begin{definition} 580Let$J_0(N)$and$J_1(N)$be the Jacobians of the Riemann surfaces$X_0(N)$and$X_1(N)$, respectively. 581\end{definition} 582 583And now we can see, at least etymologically, where the following definition comes from: 584 585\begin{definition}\label{modular} 586An abelian variety$A$is \emph{modular} if it is the quotient of$J_1(N)$for some positive integer$N$, that is, if there is a surjective map$J_1(N)\twoheadrightarrow A$. The smallest integer$N$for which this holds is referred to as the \emph{level} of the variety. 587\end{definition} 588 589\begin{remark} 590Every elliptic curve is modular. This result was conjectured by Taniyama and Shimura and has been established as an extension of the celebrated work of Wiles and Taylor-Wiles (see \cite{Taylor}). 591\end{remark} 592 593\begin{remark} 594The formal groups machinery that we established earlier for elliptic curves works on modular abelian varieties as well, although there is no general formula to work with. But one can establish that the reduction map acting on the torsion points$A(\Q)\tor\hookrightarrow A(\F_p)\tor$is injective as well, using similar methods. We do not give the details here. 595\end{remark} 596 597\begin{remark}\label{justification} 598Since$J_1(N)$has quite large dimension, we will work entirely with$J_0(N)$in the following machinery and, later, in the computations. This is purely a pragmatic consideration, since all abelian varieties that are quotients of$J_0(N)$are also quotients of$J_1(N)$, and so are modular. We should be aware, however, that there exist abelian varieties that are quotients of the latter but not the former; these will remain untreated in what follows. 599\end{remark} 600 601\begin{remark} 602Although we are only going to consider modular abelian varieties, there are many abelian varieties that are not modular. It is conjectured (in \cite{Ribet}) that an abelian variety is modular if and only if it is of $\GL_2$-type". Briefly, an abelian variety$A/\Q$is said to be of$\GL_2$-type if there exists$K \subset \Q\otimes \End_\Q(A)$for which$[K:\Q]=\dim(A)$. Here,$\End_\Q(A)$is the ring of endomorphisms of$A$over$\Q$. This characterization is a generalization of the Taniyama-Shimura conjecture, since an elliptic curve is a dimension 1 abelian variety, and taking$K=\Q$in the above definition we see that it is automatically of$\GL_2$-type. 603\end{remark} 604 605\subsection{Hecke Operators and Modular Symbols} 606 607>From here on, the theory will explicitly involve the modular subgroup$\Gamma_0(N)$and its associated$X_0(N)$and$J_0(N)$, because of a preceding remark (\ref{justification}). 608 609\begin{definition} 610The space of \emph{modular symbols} is a shorthand for writing elements of the homology group. We write$\{\alpha,\beta\}\in H_1(X_0(N),\Q)$for the homology class represented by integration from$\alpha$to$\beta$, both elements of the set of cusps$\p^1(\Q)$. 611\end{definition} 612 613\begin{remark} 614For any pair of elements$\alpha, \beta \in \p^1(\Q)$, the element$\{\alpha, \beta\}$is in fact an element of the rational homology, as mentioned above; this is a theorem of Manin and Drinfeld. 615\end{remark} 616 617Since elements of$\Gamma_0(N)$act on$\h^*$, they should also act on the modular symbols by$g(\{\alpha,\beta\})=\{g\alpha,g\beta\}$. However, for$g\in\Gamma_0(N)$, this does nothing to the homology class, because it amounts only to a change of variables in the integration. This suggests that we should formalize the behavior of modular symbols, using what we already know of the structure of$H_1(X_0(N),\Q)$. 618 619\begin{proposition} 620The modular symbols$\{\alpha,\beta\}\in H_1(X_0(N),\Q)$inherit the following properties from properties of integration: 621\begin{enumerate} 622\item 623$\{\alpha,\alpha\}=0.$624\item 625$\{\alpha,\beta\}=-\{\beta,\alpha\}.$626\item 627$\{\alpha,\beta\}+\{\beta,\gamma\}=\{\alpha,\gamma\}.$628\item\label{cov} 629As observed above, for any$g\in\Gamma_0(N)$,$\{g(\alpha),g(\beta)\}=\{\alpha,\beta\}$. 630\end{enumerate} 631\end{proposition} 632 633\begin{definition} 634Given a prime$p$not dividing the level$N$, define the \emph{Hecke operator}$T_p$acting on$H_1(X_1(N),\Q)$by 635$$T_p(\{\alpha,\beta\})=\abcd{p}{0}{0}{1}(\{\alpha,\beta\})+\sum_{k=0}^{p-1}\abcd{1}{k}{0}{p}(\{\alpha,\beta\}).$$ 636These operators commute. The ring of Hecke operators is denoted by$\T=\Z[T_1,T_2,\ldots]$, the set of$\Z$-linear combinations of the individual operators. 637\end{definition} 638 639\begin{remark} 640The individual operations$\smabcd{p}{0}{0}{1}(\{\alpha,\beta\})$and$\smabcd{1}{k}{0}{p}(\{\alpha,\beta\})$are not well-defined, in that they depend on a choice of coset representative$\{\alpha,\beta\}$. However, the sum in the definition of the Hecke operator does not. Also observe that the action" of a$\GL_2(\Z)$matrix on a modular symbol, and therefore on its constituent elements in$\p^1(\Q)$, has not yet been mentioned. By a slight abuse of confidence but not of definition, we can extend our concept of matrix action on$\p^1(\Q)$to include all elements of$\GL_2(\Z)$, since the imaginary part of an element of$\p^1(\Q)$is zero and the set is therefore preserved by linear fractional transformations. 641\end{remark} 642 643%The Hecke operator$T_n$therefore acts on the 644 645\section{Computations with Modular Abelian Varieties} 646 647Here we will briefly mention an algorithm (mostly given in \cite{Cremona}, ch. II) for finding a multiple of$\#A(\Q)\tor$for modular abelian varieties where we explicitly know the smallest level$N$of a surjective map$J_1(N)\to A$. This is essentially by way of giving background for the computations we perform in \S8. 648 649\subsection{Finding the Quotients of$J_0(N)$} 650 651The first thing to do is compute the homology group$H_1(X_0(N),\Q)$, as represented by the group of modular symbols. This group is isomorphic to a certain finite group$H(N)$with relations that are relatively easy to work with (this is a result due to Manin, cited in \cite{Cremona}, theorem 2.1.4), so the computations are easier than one would expect given the definition of the modular symbols space. The group$H(N)$decomposes into simple$\T$-modules$M_1\ldots M_{2r}$under the action of the Hecke algebra, and there is a way in which these modules pair up canonically to associate with each of the$r$abelian varieties of level$N$. 652 653\subsection{The Torsion Multiple} 654 655It is a deep fact that the characteristic polynomial of the Hecke operator$T_p$acting on the$M_i$associated to a variety, evaluated at$p+1$, gives the number of points in the reduction over$\F_p$. This is analogous to the situation with elliptic curves, although without a Weierstrass equation the Hecke operators are the only way to find the size of the reduced variety. For primes of good reduction, the torsion subgroup is injectively reduced and its order is a divisor of$\textrm{charpoly}(T_p)(p+1)$. 656 657So once we have the varieties associated to the chosen level represented as$\T$-modules$M_i$, we compute the charpoly of$T_p$acting on$M_i$, and evaluate it at$p+1$, and then take the GCD of the resulting torsion multiple for various primes$p$not dividing the level. 658 659\section{Examples} 660 661Using two computer workstations, MECCAH (the Mathematics Extreme Computation Cluster at Harvard) and Neron (named for Andr\'{e} N\'{e}ron), William Stein computed the characteristic polynomials of the first eight prime Hecke operators acting on all simple quotients of$J_0(N)$up to$N=5000$, with less comprehensive calculations up to$N=7244$, as well as the dimensions of these varieties. We used this data to compute a multiple of the torsion group for each one, as above, using primes up to 19. 662 663This first table shows the possible torsion multiples, sorted by dimension, for low dimensions. Full ranges are given by a dash.\\ 664 665\begin{tabular}{|c|c|} 666\hline 667Dimension & Torsion Multiples Found \\ 668\hline 6691 & 1-10, 67012, 67116, 672$20^*$\\ 673\hline 6742 & 1-17, 67519, 67620, 67722-24, 67827, 67928, 68030-32, 68136, 68237, 68344, 68452, 68556, 68664\\ 687\hline 6883 & 1-20, 68922-24, 69028, 69131, 69232, 69336-38, 69449, 69558, 69664, 69772, 69880, 69983$^\dagger$, 70092, 701160\\ 702\hline 7034 & 1-17, 70419-29, 70531, 70632, 70734-36, 70838, 70940, 71044, 71148, 71249, 71355, 71456, 71564, 71674, 71776,\\& 71896, 719152, 720176, 721208, 722256, 723328$^\dagger$\\ 724\hline 7255 & 1-14, 72616, 72718-25, 72827-29, 72931, 73032, 73134-36, 73238, 73341$^\dagger$, 73444, 73548, 73650, 73754, 73856,\\& 73958, 74063, 74164, 74268, 74372, 74486$^\dagger$, 74592, 746104, 747110, 748128, 749212$^\dagger$, 750224, 751272, 752288, 753320\\ 754\hline 7556 & 1-18, 75620, 75722, 75824, 75925, 76027, 76128, 76231, 76332, 76434, 76536, 76637, 76740, 76841$^\dagger$, 76948, 77049,\\& 77152, 77253$^\dagger$, 77356, 77464, 77570, 77674, 77776, 77880, 77996, 780104, 781120, 782124, 783128, 784192, 785212$^\dagger$,\\& 786248, 787288, 788296, 789432\\ 790\hline 791\end{tabular}\\ 792 793Two things jump out at us. First, the starred number in the above chart is the only case where the torsion multiple is immediately seen to be too large, by Theorem \ref{Mazur}. The variety is the tenth in level 2310. Since$2310=2\cdot3\cdot5\cdot7\cdot11$, it is likely that we merely didn't use enough primes (the computations could only take advantage of the primes 13, 17, and 19). A quick follow-up search in the modular forms database shows that our surmise was correct: using primes through 53, the GCD of the torsion multiples is reduced quite drastically, to 4, which is well within the bound imposed by Theorem \ref{Mazur}. 794 795Next, the daggers in the table represent torsion multiples divisible by a prime greater than 40. Since there isn't even any 11-torsion in dimension 1, it's interesting to investigate some examples of torsion points of large prime order in higher dimensions. We'll make a separate table of these$\dagger$-ed varieties; where possible we use the modular forms database to see whether the torsion multiple gets smaller using primes up to 53.\\ 796 797\begin{tabular}{|c|c|c|c|c|} 798\hline 799N &Number & Dimension & Multiple of & Multiple of torsion\\ 800 (level) & within level & & torsion & (using MFD) \\ 801\hline 8025010 & 16 & 3 & 83 & --\\ 803\hline 804498 &7 &4 &328 & 82\\ 805\hline 806830 &11 &5 &41 & 41\\ 807\hline 8081038 &10 &5 &86 & 86\\ 809\hline 810642 &8 &5 &212& 106\\ 811\hline 81283 &2 &6 &41& 41$^\ddagger$\\ 813\hline 814326 &5 &6 &41& 41$^\ddagger$\\ 815\hline 8162158 &10 &6 &41& 41\\ 817\hline 8181070 &12 &6 &53 &53\\ 819\hline 820422 &6 &6 &212 & 53$^\ddagger$\\ 821\hline 822\end{tabular}\\ 823 824Let us consider the second entry in the table: the 7th modular abelian variety of level 498; it has dimension 4. Using primes up to 19, we computed 328 as the torsion multiple and 40709 (a prime) as the multiple of the field discriminant. Since 498 is only divisible by two of the primes on our list (2 and 3), one may be inclined to trust," more or less, the computation of the torsion multiple. In particular, 328 has a large prime factor, namely 41, and we'd like to know whether we can observe 41-torsion in the variety explicitly. 825 826There is a series of maps 827$$X_0(N) \hookrightarrow J_0(N) \twoheadrightarrow A$$ 828where the first map between points and divisors is$P \mapsto [P-\infty]$. The image of the cusps in$X_0(N)$under this map can be used to generate a subgroup of the Jacobian called the cuspidal subgroup,$C$. It is a fact that$C(\Q)\subset J_0(N)(\Q)\tor$. Then under the quotient map into$A(\Q)$it will end up as a subgroup of the torsion on the variety. Thus this is a way of explicitly constructing a lower bound on the torsion, by actually finding a subgroup of it. 829 830So the natural thing to do is to try that with this variety. We construct it using MAGMA, as follows: 831\begin{verbatim} 832> S := CuspidalSubspace(ModularSymbols(498,2,+1)); 833> D := TraceSortDecomposition(NewformDecomposition(S)); 834> A := D[15]; 835\end{verbatim} 836Then the command \verb+RationalCuspidalSubgroup(A)+ gives the subgroup generated by the image of the cuspidal subgroup under quotient. In this case, it returns \verb+Abelian Group of order 1+. 837 838(Note that in the varieties marked with a$\ddagger$in the table above, this method produces a nontrivial subgroup and therefore confirms the high prime torsion explicitly.) 839 840Perhaps unfortunately, in the example we were considering, the subgroup of the torsion group that this method produces is trivial. But there does seem to be 41-torsion in the variety. If we use primes up to 53 (computed in the modular forms database), the torsion multiple is refined to 82. All this is very suggestive, but we'd like to know whether 41 isn't just a red herring that keeps appearing. 841 842To this end, we can turn to the Birch and Swinnerton-Dyer conjecture to make predictions. The conjecture supposes an equality 843$$\frac{L(A,1)}{\Omega_A} = \frac{\prod_{p|N} c_p \cdot \#\Sha(A)}{\#A(\Q)\tor\cdot\#A^\vee(\Q)\tor},$$ 844where the left hand side is an analytic invariant associated to the variety and the right hand side contains certain algebraic invariants. The ones we are particularly concerned with are the torsion group sizes in the denominator on the right, of course. In the example above with level 498, many of these quantities can be computed explicitly. Using the MAGMA commands \verb+LRatio+ and \verb+TamagawaNumber+ (the former predicts the left hand side of the equation up to a constant conjecturally equal to 1), we find that the BSD conjecture predicts that 845$$56=\frac{(2296\cdot 164\cdot 1)\#\Sha(A)}{\#A(\Q)\tor\cdot\#A^\vee(\Q)\tor}.$$ 846All the remaining unknown quantities are integers, and the numerator on the right hand side is divisible by$41^2$, while the left hand side has no divisors of 41. Hence if BSD is correct, we should expect the variety and its dual to have 41-torsion. (We expect both of them to have it because neither of them can have two factors of 41, since the torsion multiple is an upper bound on all curves in the isogeny class.) Often we can use the BSD conjecture to make statements about the torsion subgroup. 847 848\subsection{Bounds using Good Reduction} 849 850If$A$is of dimension 2 and has good reduction at the prime 3, then$\#A(\Q)\tor\le\#A(\F_3)$. It is known that the characteristic polynomial of the Hecke operator$T_3$factors as$(x-a_3)(x-a_3^\sigma)$. Further, a result due to Deligne states that$|a_p|,|a_p^\sigma| \le 2p^{\frac{k-1}{2}}$, where$k$is the dimension. In this case,$a_3$and its conjugate are bounded in absolute value by$2\sqrt{3}$. Writing these as elements of the ring of integers in the real quadratic field$\Q[\sqrt{D}]$, we have$|a+b\sqrt{D}|,|a-b\sqrt{D}|\le 2\sqrt{3}$. We'd like to bound the possible value of the characteristic polynomial evaluated at 4=3+1, which would tell us the number of points over$\F_3$. 851 852Plugging in$x=4$and multiplying out, we need to find the maximal value of$16 + (a^2-b^2D) -8a$, which is at its largest when$b=0$and, if the absolute value of$a$is bounded, when$a<0is minimized within the range. We do some fiddling with Deligne bound and the triangle inequality to find this range: 853\begin{align*} 854|2a|&\le|a+b\sqrt{D}|+|a-b\sqrt{D}|\\ 855|2a|&\le4\sqrt{3}\\ 856|a|&\le2\sqrt{3} 857\end{align*} 858Sincea$can only take on discrete integer or possibly half-integer values,$a=-3$is the least possible; and thus the number of torsion points on a dimension 2 modular abelian variety with good reduction at$p=3$is at most$(4-(-3))^2=49$. 859 860To confirm, we search our data for varieties with dimension 2 and a computed torsion multiple greater than 49:\\ 861 862\begin{tabular}{|c|c|c|c|c|} 863\hline 864N (level) &Number within & Dimension & Multiple of field& Multiple of\\ 865 & level & & discriminant & torsion \\ 866\hline 867390 &8 &2 &32 &56\\ 868\hline 8692574 &26 &2 &164 &64\\ 870\hline 8713120 &29 &2 &32 &56\\ 872\hline 8734368 &32 &2 &17 &52\\ 874\hline 8755070 &29 &2 &32 &56\\ 876\hline 8775850 &64 &2 &32 &56\\ 878\hline 879\end{tabular}\\ 880 881Happily enough, all the levels here are divisible by 3, so we don't have to worry about the torsion multiple violating our bound. In fact, for a curve with good reduction at 3, the highest torsion multiple in the data we have computed so far is 22. The variety in which this putative torsion occurs is a quotient of$J_0(322)$. As in the previous section, for this variety \verb+RationalCuspidalSubgroup+ is trivial, but the BSD conjecture suggests that there's at least 11-torsion in the variety and its dual. 882 883This table also serves to illustrate two contradictory effects, since the torsion multiple is high and many small primes divide the level. First, since many small primes divide the level, the torsion multiple may be too high, since we have fewer numbers in the GCD calculation. Second, if the torsion is high, it ought to have many small primes dividing the level, so as to have bad reduction at the small primes and therefore the capacity for higher torsion. It is not easy to tell which effect is dominant in a particular example. 884 885\subsection{Conjectures} 886 887Sifting through the data, we find strong evidence for the following conjectures. 888 889\begin{conjecture}\label{conj1} 890\begin{enumerate} 891\item 892Given$p_1,p_2,p_3$distinct primes, if the numerator of$\frac{p_i-1}{12}$, which we'll call$\ell_i$, is a prime for some$i$, then there is an$\ell_i$-torsion point in some modular abelian variety whose level is$p_1\cdot p_2\cdot p_3$. 893\item 894An analogous statement holds for any odd number of distinct primes$p_i$. 895\item 896The above statements hold for any prime$\ell_i$dividing the numerator of$\frac{p_i-1}{12}$. 897\end{enumerate} 898\end{conjecture} 899 900\begin{conjecture} 901For any prime$p>2$, there is a$p$-torsion point in some modular abelian variety whose level is$p^3$. 902\end{conjecture} 903 904For an example of the first statement, recall the abelian variety with level$498=2\cdot3\cdot83$above; the BSD computation suggested strongly that it had a 41-torsion point. Similar computations can be made with other varieties in that table: the one with level$642=2\cdot3\cdot107$appeared to have a 53-torsion point. In fact, upon closer inspection, all the varieties with high torsion that we found seem to satisfy this conjecture. 905 906This first observation is bound to be related to the following theorem, which is proved in \cite{Mazur}: 907 908\begin{theorem}[Mazur] 909Let$p>3$be a prime, and let$n$be the numerator of$\frac{p-1}{12}$. Then the torsion subgroup of the Mordell-Weil group of$J_0(p)$is a cyclic subgroup of order$n$. 910\end{theorem} 911 912Although this theorem addresses the case of prime level, it is not immediately clear why Conjecture \ref{conj1} seems to hold when the level is the product of an odd number of distinct primes. 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